Question

Prove that √5 is irrational

Answer 1

Hi there!

Assume that √5 is rational

√5 = p/q   where p and q are coprime

P = √5q

Squaring both sides

P² =  (√5q)²
P² = 5q² ----(1)

Now p² is divisible by 5 so p is also divisible by 5

So,     p = 5r        (where r is any positive integer) ----(2)

Putting value of eq. (2) in (1) we get,

25r² = 5q²

Now on dividing from 5 on both sides we get,

5r² = q²

So we can conclude that p and q both have common factor 5 so they are not co-prime.

This problem arised due to wrong assumption that √5 is rational.

So, √5 is irrational.

Cheers!

Answer 2

$\huge \red{\bf Answer}$

We need to prove that √5 is irrational

$\pink{\rm Proof:}$

Let us assume that √5 is a rational number.

Sp it t can be expressed in the form p/q where p,q are co-prime integers and q≠0

⇒√5=p/q

On squaring both the sides we get,

⇒5=p²/q²

⇒5q²=p² —————–(i)

p²/5= q²

So 5 divides p

ANSWER

p is a multiple of 5

⇒p=5m

⇒p²=25m² ————-(ii)

From equations (i) and (ii), we get,

5q²=25m²

⇒q²=5m²

⇒q² is a multiple of 5

⇒q is a multiple of 5

Hence, p,q have a common factor 5. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

$\green{ \bf \sqrt{5} \: is \: a \: irrational \: number}$